# Logarithm of Infinite Product of Real Numbers

## Theorem

Let $(a_n)$ be a sequence of strictly positive real numbers.

### Convergence

The following statements are equivalent:

$(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
$(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.

### Divergence to zero

The following are equivalent:

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.
The series $\ds \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.

### Divergence to infinity

The following statements are equivalent::

$(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.
$(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$.